Frequently/sometimes. And common availability has nothing to it. I choose mounts according to class and my general "feel" how certain mount suits it. Class and its mount has to go togther. For example - my Druid has Flight Form, no real need to have a mount. But during late BC I started farming both rep and gold for a Cenarion War Hippogryph. Finished at the start of WotLK (still remember doing almost 2 lvls on those ears in Borean Tundra). CWH is my Druid's mount ever since. And I got Lunarwing Form because I use it and I like it. For short flights I prefer form (insta cast), for longer ones I use mount.
Another example is Warlock - I managed to snatch Headless Horseman mount for my Affliction (while using default Dreadsteed as Demo). Warlock's Class mount will be nice alternative. In the end I'll be probably using all class mounts rather frequently, Paladin one for sure as default since I still miss Charger on my Draenei. Those mounts are well designed and really into-class feel.
Not at all. I consider myself a rare mount collector. If I like it I'll probably farm it just to add it to my collection. So far i was lucku enough to get almost all I wanted, the ones I'm missing are not desired much but if I get them they would be nice addition.
And your 2nd question is partially true in certain cases. Paladin, Rogue and Demon Hunter for sure. My Draenei Paladin will finally get paladin-ish horse back, Rogue will get fitting Raven and Demon Hunter gets perfectly suiting Fel Bat. Those three classes have mounts I'll ever need in the foreseeable future. The rest? Druid with her shapeshifting can easily use both (new form and my old mount), Priest will have an option (Dark Phoenix or this new Owl). Overall class mounts will introduce nice additional option. Only Warrior is unimpressed.